The ability of a radar to precisely locate a target is limited by the beamwidth of the radar, since a radar return can come from anywhere in the cone formed by the beam. By making the beam as narrow as possible, the target can be located more precisely. However, physics dictates that in order to make the beam narrower, the physical size of the antenna must correspondingly increase. Thus, to precisely locate a target with a radar requires a large antenna.
One way to escape from this physics-induced tradeoff is to use Interferometric methods, in which two or more radar receivers are used instead of the normal one. These receivers are placed a distance apart, and by comparing the differences in phase between the received signals, it is possible to obtain a much more accurate location of a target than with a single receiver. If there are enough receivers, it may be possible to localize the target without ambiguity. However, if we use only two receivers, or in certain geometries with more than two receivers, physics again limits the applicability of this approach. Because the phase can be measured only modulo 2π, the location of the target cannot be determined uniquely because of phase ambiguity. As a result, the target can be in any of a small number of discrete locations, each contained within the radar beam, and each located more precisely than is possible without the interferometry. The tradeoff with the Interferometric approach is between the precision with which we can locate the target, the number of receivers (and antennas required), and the introduction of a number of “ghost” targets. From a single radar return and using two receivers, there is no way to differentiate the true target from the ghost targets that arise from the phase ambiguities. In order to differentiate between the true target and the ghost targets, multiple radar returns must be utilized.
The use of two or more receivers does have an additional drawback, in that the antenna design becomes more complicated. At worst, two or more separate antennas will be required; at best, one single antenna may be shared by the two or more receivers. However, with interferometry it is possible to achieve performance with two antennas (and two receivers) that is much superior to that achieved with a single receiver and a single antenna of twice the size.
As shown in FIG. 1 the basic geometry of an Interferometric radar with two receivers 10, 20 is shown. Other cases, with more receivers or different alignments, may be used. The two receivers 10, 20 are displaced from a single transmitter by distances of λN1 and λN2, with λ being the wavelength of the radar. The signal paths from the transmitter to the target and back to the two receivers 10, 20 have slightly different paths. If R is the distance from the transmitter to the receiver, then the length of the signal path to the receiver 10 is given, to first order in λN1/R by:
                              d          1                =                              2            ⁢            R                    +                      λ            ⁢                                                  ⁢                          N              1                        ⁢                                                                                (                                                                  x                        t                                            -                                              x                        0                                                              )                                    ⁢                  sin                  ⁢                                                                          ⁢                  θ                                -                                                      (                                                                  z                        t                                            -                                              z                        0                                                              )                                    ⁢                  cos                  ⁢                                                                          ⁢                  θ                                            R                                                          (        1        )            the difference in path length for the two receivers 10, 20 is then
                    Δ        =                              λ            ⁡                          (                                                N                  1                                +                                  N                  2                                            )                                ⁢                                                                      (                                                            x                      t                                        -                                          x                      0                                                        )                                ⁢                sin                ⁢                                                                  ⁢                θ                            -                                                (                                                            z                      t                                        -                                          z                      0                                                        )                                ⁢                cos                ⁢                                                                  ⁢                θ                                      R                                              (        2        )            and the difference in phase of signals arriving at the two receivers 10, 20 is just
                    Φ        =                  2          ⁢                      π            ⁡                          (                                                N                  1                                +                                  N                  2                                            )                                ⁢                                                                      (                                                            x                      t                                        -                                          x                      0                                                        )                                ⁢                sin                ⁢                                                                  ⁢                θ                            -                                                (                                                            z                      t                                        -                                          z                      0                                                        )                                ⁢                cos                ⁢                                                                  ⁢                θ                                      R                                              (        3        )            
Let Δθ and ΔΦ be the difference in elevation and azimuth, respectively, between the target position and the direction of the radar beam. Then the phase difference isΦ=2π(N1+N2)cos(ΔØ)sin(Δθ)≈2π(N1+N2)Δθ  (4)where we assume that the target is inside the radar beam and the small angle approximation is valid. |Δθ| must be less than the vertical radar half-beamwidth.
Suppose now a measurement has been made, resulting in a measured phase difference of Φm±δΦ. The measured phase difference is ambiguous. A measurement of Φm means only that the true phase difference is Φm+2nπ, where n is any integer. Hence, solving Eq. (4) for Δθ, the following is determined:
                    Δθ        =                                            Φ              m                        +            δΦ            +                          2              ⁢                                                          ⁢              n              ⁢                                                          ⁢              π                                            2            ⁢                          π              ⁡                              (                                                      N                    1                                    +                                      N                    2                                                  )                                                                        (        5        )            
The possible values for n are limited only by the requirement that |Δθ| be less than the radar beamwidth. Only one of the possible values of n represents the target; the other values are ghost targets.
Noteworthy is the dependence of the calculation of Δθ on δΦ, as this determines how accurately one can localize the target. Increasing the separation between the two receivers 10, 20 increases the accuracy, but at a cost of increasing the number of ghosts possible. As an example, consider an example of a 94 GHz Interferometric radar to obtain some indication of the increase in capability. For this radar, assume that the beamwidth in the vertical direction is 4°, and the nominal separation between the two receivers 10, 20 is 37 wavelengths. Assume also that the expected resolution of the phase measurement is ±5°; then the nominal resolution of the Interferometric radar will be ±0.02°. There will be as many as three ghosts within the 4° beam, in addition to the real target. The resulting resolution is 20 times better using interferometry. Note that to achieve this kind of accuracy with a single receiver would require an antenna 20 times larger.